Method for determining power limit of battery, and battery management system

ABSTRACT

Disclosed is a method and a battery management system for periodically determining at least one of charge power limit and discharge power limit of a battery. The method according to an embodiment of the present disclosure includes predicting the terminal voltage and the net resistance of the battery after a predefined time from the current time using an equivalent circuit model for the battery, and determining the charge power limit or the discharge power limit of the battery from the predicted terminal voltage and net resistance.

TECHNICAL FIELD

The present disclosure relates to a method and a battery managementsystem for periodically determining at least one of charge power limitand discharge power limit of a battery.

The present application claims priority to Korean Patent Application No.10-2018-0013014 filed in the Republic of Korea on Feb. 1, 2018, thedisclosure of which is incorporated herein by reference.

BACKGROUND ART

Recently, there is dramatically growing demand for portable electronicproducts such as laptop computers, video cameras and mobile phones, andwith the extensive development of electric vehicles, accumulators forenergy storage, robots and satellites, many studies are being made onhigh performance batteries that can be recharged repeatedly.

Currently, commercially available batteries include nickel-cadmiumbatteries, nickel-hydrogen batteries, nickel-zinc batteries, lithiumbatteries and the like, and among them, lithium batteries have little orno memory effect, and thus they are gaining more attention thannickel-based batteries for their advantages of free charging anddischarging, a very low self-discharge rate and high energy density.

To safely use a battery, it is necessary to determine the power limit ofthe battery separately for the charging process and the dischargingprocess, and control based on the determined power limit. To determinethe power limit of the battery, it is necessary to predict the terminalvoltage of the battery in advance.

However, because in some instances, the battery works in an environmentin which the terminal voltage and the current fluctuate so much, it isnot easy to predict how much the terminal voltage of the battery will bein a predetermined time from the current time. If the power limit of thebattery is determined based on incorrectly predicted terminal voltage,the safety of the battery reduces.

DISCLOSURE Technical Problem

The present disclosure is designed to solve the above-described problem,and therefore the present disclosure is directed to providing a methodand a battery management system that predicts the terminal voltage andthe net resistance of a battery at a predetermined time from the currenttime using an equivalent circuit model for the battery, and periodicallyupdates the charge power limit or the discharge power limit of thebattery from the predicted terminal voltage and net resistance.

These and other objects and advantages of the present disclosure can beunderstood by the following description and will be apparent from theembodiments of the present disclosure. Further, it will be readilyunderstood that the objects and advantages of the present disclosure canbe realized by the means set forth in the appended claims andcombinations thereof.

Technical Solution

Various embodiments of the present disclosure for achieving theabove-described object are as follows.

A method according to an embodiment of the present disclosure is fordetermining the power limit of a battery using an equivalent circuitmodel for the battery. The equivalent circuit model includes a firstresistor, a second resistor connected in series to the first resistorand a capacitor connected in parallel to the second resistor. The methodincludes measuring a terminal voltage and a current of the battery,estimating a resistance of the first resistor, a resistance of thesecond resistor and a polarization voltage of the battery, predicting aterminal voltage of the battery after a predefined time based on theresistance of the second resistor, the measured terminal voltage, themeasured current and the polarization voltage, predicting a netresistance of the battery after the predefined time based on theresistance of the first resistor and the resistance of the secondresistor, determining a charge current limit based on the measuredcurrent, the predicted terminal voltage, the predicted net resistanceand a given maximum charge voltage, determining a charge voltage limitbased on the measured terminal voltage, the measured current, the chargecurrent limit and the resistance of the first resistor, and determininga charge power limit of the battery based on the charge current limitand the charge voltage limit.

The method may further include determining a discharge current limitbased on the measured current, the predicted terminal voltage, thepredicted net resistance and a given minimum discharge voltage,determining a discharge voltage limit based on the predicted terminalvoltage, the measured current, the discharge current limit and the netresistance, and determining a discharge power limit of the battery basedon the discharge current limit and the discharge voltage limit.

In the step of predicting the terminal voltage of the battery after thepredefined time, the following Equation 1 may be used.

$\begin{matrix}{{V_{pred}\;(n)} = {{V(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}\left\{ {{{R_{2{\_ est}}(n)}{I(n)}} - {V_{pola}(n)}} \right\}}}} & \left\langle {{Equation}\mspace{14mu} 1} \right\rangle\end{matrix}$

wherein V(n) is the measured terminal voltage, R_(2_est)(n) is theestimated resistance of the second resistor, I(n) is the measuredcurrent, V_(pola)(n) is the polarization voltage, T_(hard) is thepredefined time, and V_(pred)(n) is the predicted terminal voltage.

In the step of predicting the net resistance of the battery in thepredefined time, the following Equation 2 may be used.

$\begin{matrix}{{R_{net}(n)} = {{R_{1{\_ est}}(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}{R_{2{\_ est}}(n)}}}} & \left\langle {{Equation}\mspace{14mu} 2} \right\rangle\end{matrix}$

wherein R_(1_est)(n) is the estimated resistance of the first resistor,R_(2_est)(n) is the estimated resistance of the second resistor,T_(hard) is the predefined time, and R_(net)(n) is the predicted netresistance.

In the step of determining the charge current limit, the followingEquation 3 may be used.

$\begin{matrix}{{I_{{limit}\_ c}(n)} = {{I(n)} + \frac{V_{\max} - {V_{pred}(n)}}{R_{net}(n)}}} & \text{〈Equation~~3〉}\end{matrix}$

wherein I(n) is the measured current, R_(net)(n) is the predicted netresistance, V_(pred)(n) is the predicted terminal voltage, V_(max) isthe given maximum charge voltage, and I_(limit_c)(n) is the chargecurrent limit.

In the step of determining the charge voltage limit, the followingEquation 4 may be used.V _(limit_c)(n)=V(n)+{I _(limit_c)(n)−I(n)}R _(1_set)(n)  <Equation 4>

wherein V(n) is the measured terminal voltage, I(n) is the measuredcurrent, I_(limit_c)(n) is the charge current limit, R_(1_set)(n) is theestimated resistance of the first resistor, and V_(limit_c)(n) is thecharge voltage limit.

In the step of determining the discharge current limit, the followingEquation 5 may be used.

$\begin{matrix}{{I_{{limit}\_ d}(n)} = {{I(n)} + \frac{V_{\min} - {V_{pred}(n)}}{R_{net}(n)}}} & \text{〈Equation~~5〉}\end{matrix}$

wherein I(n) is the measured current, R_(net)(n) is the predicted netresistance, V_(pred)(n) is the predicted terminal voltage, V_(min) isthe given minimum charge voltage, and I_(limit_d)(n) is the dischargecurrent limit.

In the step of determining the discharge voltage limit, the followingEquation 6 may be used.V _(limit_d)(n)=V _(pred)(n)+{I _(limit_d)(n)−I(n)}R_(net)(n)  <Equation 6>

wherein V_(pred)(n) is the predicted terminal voltage, I(n) is themeasured current, I_(limit_d)(n) is the discharge current limit,R_(net)(n) is the predicted net resistance, and V_(limit_d)(n) is thedischarge voltage limit.

A battery management system according to another embodiment of thepresent disclosure determines the power limit using the equivalentcircuit model. The battery management system includes a sensing unitconfigured to measure a terminal voltage and a current of the battery,and a control unit operably coupled to the sensing unit. The controlunit estimates a resistance of the first resistor, a resistance of thesecond resistor and a polarization voltage of the battery. The controlunit predicts a terminal voltage of the battery after a predefined timebased on the resistance of the second resistor, the measured terminalvoltage, the measured current and the polarization voltage. The controlunit predicts a net resistance of the battery after the predefined timebased on the resistance of the first resistor and the resistance of thesecond resistor. The control unit determines a charge current limitbased on the measured current, the predicted terminal voltage, thepredicted net resistance and the given maximum charge voltage. Thecontrol unit determines a charge voltage limit based on the measuredterminal voltage, the measured current, the charge current limit and theresistance of the first resistor. The control unit determines a chargepower limit of the battery based on the charge current limit and thecharge voltage limit.

The control unit may determine a discharge current limit based on themeasured current, the predicted terminal voltage, the predicted netresistance and the given minimum discharge voltage. The control unit maydetermine a discharge voltage limit based on the predicted terminalvoltage, the measured current, the discharge current limit and the netresistance. The control unit may determine a discharge power limit ofthe battery based on the discharge current limit and the dischargevoltage limit.

Advantageous Effects

According to at least one of the embodiments of the present disclosure,when charging a battery, it is possible to predict the terminal voltageand the net resistance of the battery at a predetermined time from thecurrent time using an equivalent circuit model for the battery, andperiodically update the charge power limit of the battery from thepredicted terminal voltage and net resistance.

Additionally, according to at least one of the embodiments of thepresent disclosure, when discharging a battery, it is possible topredict the terminal voltage and the net resistance of the battery at apredetermined time from the current time using an equivalent circuitmodel for the battery, and periodically update the discharge power limitof the battery from the predicted terminal voltage and net resistance.

The effects of the present disclosure are not limited to the effectsmentioned above, and these and other effects will be clearly understoodby those skilled in the art from the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings illustrate a preferred embodiment of thepresent disclosure, and together with the detailed description of thepresent disclosure described below, serve to provide a furtherunderstanding of the technical aspects of the present disclosure, andthus the present disclosure should not be construed as being limited tothe drawings.

FIG. 1 is a diagram showing the functional configuration of a batterypack according to an embodiment of the present disclosure.

FIG. 2 is a diagram showing an exemplary equivalent circuit model of abattery.

FIG. 3 is a flowchart showing a method for estimating the resistance ofa first resistor as one of parameters of an equivalent circuit modelaccording to an embodiment of the present disclosure.

FIG. 4 shows graphs for reference in describing the method of FIG. 3.

FIGS. 5 and 6 are flowcharts showing a method for estimating theresistance of a second resistor as one of parameters of an equivalentcircuit model according to an embodiment of the present disclosure.

FIG. 7 shows a graph for reference in describing the method of FIG. 5.

FIG. 8 is a flowchart showing a method for determining the charge powerlimit and the discharge power limit of a battery according to anotherembodiment of the present disclosure.

FIGS. 9 and 10 show different exemplary graphs for reference indescribing the method of FIG. 8.

MODE FOR DISCLOSURE

Hereinafter, the preferred embodiments of the present disclosure will bedescribed in detail with reference to the accompanying drawings. Priorto the description, it should be understood that the terms or words usedin the specification and the appended claims should not be construed asbeing limited to general and dictionary meanings, but interpreted basedon the meanings and concepts corresponding to the technical aspects ofthe present disclosure on the basis of the principle that the inventoris allowed to define the terms appropriately for the best explanation.

Therefore, the embodiments described herein and illustrations shown inthe drawings are just a most preferred embodiment of the presentdisclosure, but not intended to fully describe the technical aspects ofthe present disclosure, so it should be understood that a variety ofother equivalents and modifications could be made thereto at the time offiling the application.

Additionally, in describing the present disclosure, when it is deemedthat a certain detailed description of relevant known elements orfunctions renders the key subject matter of the present disclosureambiguous, the detailed description is omitted herein.

The terms including the ordinal number such as “first”, “second” and thelike, are used to distinguish one element from another among variouselements, but not intended to limit the elements by the terms.

Unless the context clearly indicates otherwise, it will be understoodthat the term “comprises” or “includes” when used in this specification,specifies the presence of stated elements, but does not preclude thepresence or addition of one or more other elements. Additionally, theterm <control unit> as used herein refers to a processing unit of atleast one function or operation, and this may be implemented by hardwareor software alone or in combination.

In addition, throughout the specification, it will be further understoodthat when an element is referred to as being “connected to” anotherelement, it can be directly connected to the other element orintervening elements may be present.

FIG. 1 is a diagram showing the functional configuration of a batterypack 1 according to an embodiment of the present disclosure.

Referring to FIG. 1, the battery pack 1 includes a battery 10, a switch20 and a battery management system 100. The switch 20 is configured toadjust the magnitude of charge current and/or discharge current of thebattery 10 in response to a switching signal (for example, a pulse widthmodulation signal) from the battery management system 100.

The battery management system 100 is electrically coupled to the battery10 and configured to monitor and control the state of the battery 10.The battery management system 100 includes a sensing unit 110, a memory120, a control unit 130 and a communication interface 140.

The sensing unit 110 includes a current measuring unit 111. The currentmeasuring unit 111 measures the current of the battery 10 at each timestep defined by a predefined length of time, and transmits a currentsignal indicating the measured current to the control unit 130. Thecurrent at the time of discharging the battery 10 may be referred to as‘discharge current’, and the current at the time of charging the battery10 may be referred to as ‘charge current’. The control unit 130 mayconvert the current signal in analog form transmitted from the currentmeasuring unit 111 to current data in digital form. Hereinafter, assumethat the current at the time of charging is measured as a positivevalue, and the current at the time of discharging is measured as anegative value.

The sensing unit 110 may further include a voltage measuring unit 112.The voltage measuring unit 112 measures the terminal voltage of thebattery 10 at each time step, and transmits a voltage signal indicatingthe measured terminal voltage to the control unit 130. The control unit130 may convert the voltage signal in analog form transmitted from thevoltage measuring unit 112 to voltage data in digital form.

The sensing unit 110 may further include a temperature measuring unit113. The temperature measuring unit 113 measures the temperature of thebattery 10 at each time step, and transmits a temperature signalindicating the measured temperature to the control unit 130. The controlunit 130 may convert the temperature signal in analog form transmittedfrom the temperature measuring unit 113 to temperature data in digitalform. The current measuring unit 111, the voltage measuring unit 112 andthe temperature measuring unit 113 may operate in time synchronizationwith each other. Hereinafter, k^(th) time step is expressed as ‘timestep k’. Additionally, the terminal voltage and the current measured bythe sensing unit 110 at the time step k are respectively expressed asV(k) and I(k).

The memory 120 may additionally store data, instructions and softwarerequired for the overall operation of the battery management system 100.The memory 120 may store data indicating the result of the operationperformed by the control unit 130. The terminal voltage, the currentand/or the temperature of the battery 10 measured by the sensing unit110 at each time step may be recorded in the memory 120 in a sequentialorder. The memory 120 may include at least one type of storage medium offlash memory type, hard disk type, Solid State Disk (SSD) type, SiliconDisk Drive (SDD) type, multimedia card micro type, random access memory(RAM), static random access memory (SRAM), read-only memory (ROM),electrically erasable programmable read-only memory (EEPROM) andprogrammable read-only memory (PROM).

The control unit 130 is operably coupled to the sensing unit 110, thememory 120 and the communication interface 140. The control unit 130records the terminal voltage, the current and/or the temperature of thebattery 10 measured by the sensing unit 110 at each time step in thememory 120 in a sequential order. The control unit 130 may move, at eachtime step, a sliding time window having a predefined size as much as atime interval Δt of the time step, and read a plurality of terminalvoltages and a plurality of currents measured in the sliding time windowamong all terminal voltages and currents recorded in the memory 120 fromthe memory 120. For example, when the time interval of the time step is0.01 sec, and the size of the sliding time window is 10 sec, 1000terminal voltages and 1000 currents may be read from the memory 120 ateach time step.

The control unit 130 may be physically implemented using at least one ofapplication specific integrated circuits (ASICs), digital signalprocessors (DSPs), digital signal processing devices (DSPDs),programmable logic devices (PLDs), field programmable gate arrays(FPGAs), microprocessors and electrical units for performing otherfunctions.

The communication interface 140 may be coupled to an external device 2such as an electronic control unit (ECU) of an electric vehicle toenable communication between. The communication interface 140 mayreceive a command message from the external device 2, and provide thereceived command message to the control unit 130. The command messagemay be a message that requests the activation of a specific function ofthe apparatus. The communication interface 140 may transmit anotification message from the control unit 130 to the external device 2.The notification message may be a message for informing the externaldevice 2 of the result of the function (for example, the state of chargeof the battery) performed by the control unit 130.

FIG. 2 is a diagram showing an exemplary equivalent circuit model 200 ofthe battery.

Referring to FIG. 2, the equivalent circuit model 200 may include avoltage source 205, a first resistor 210, a second resistor 220 and acapacitor 230. The parameters of the equivalent circuit model 200 mayinclude the resistance of the first resistor 210, the resistance of thesecond resistor 220 and the capacitance of the capacitor 230.

The voltage source 205 represents an open circuit voltage (OCV) V_(OCV)of the battery determined from the State Of Charge (SOC) and thetemperature of the battery. That is, the OCV V_(OCV) may be uniquelydetermined when the SOC and the temperature are determined. The OCVV_(OCV) may be predefined for each SOC and each temperature. That is, anOCV-SOC map defining a correlation between the SOC, the temperature andthe OCV of the battery may be pre-stored in the memory 110. The OCV atk^(th) time step may be expressed as V_(OCV)(k).

The first resistor 210 models short-term voltage fluctuations by thecurrent flowing in the battery. The terminal voltage measured at thetime of charging the battery is higher than the OCV due to the internalresistance of the battery 10. On the contrary, the terminal voltagemeasured at the time of discharging the battery is lower than the OCV.

The second resistor 220 and the capacitor 230 are connected to eachother in parallel. As shown, the second resistor 220 may be connected inseries to the first resistor 210. A parallel connection circuit of thesecond resistor 220 and the capacitor 230 may be referred to as an ‘RCpair’. As opposed to the first resistor 210, the second resistor 220 isconnected in parallel to the capacitor 230. Accordingly, the RC pair maymodel the polarization voltage generated during charging and dischargingthe battery. That is, the parallel combination of the second resistor220 and the capacitor 230 is for modeling the transient response historyof the battery.

Assume that the resistance of the first resistor 210 and the resistanceof the second resistor 220 are constant as R₁ and R₂, respectively. IfΔt is very small, each of the terminal voltage and the current of thebattery 10 measured at an arbitrary time step may be constant until thenext time step, therefore the OCV of the voltage source 205 may be alsohandled as being constant between two adjacent time steps.

Assume that the polarization voltage by the RC pair at an arbitrary timepoint at which the time step k starts is V_(pola), and the resistance ofthe first resistor 210 and the resistance of the second resistor 220 areconstant as R₁ and R₂, respectively, from the time step k to the timestep q. Then, voltage V_(model)(q) of the equivalent circuit model 200at the time step q may be expressed as the following Equation 1.

$\begin{matrix}{{V_{model}(q)} = {{V_{ocv}(q)} + {R_{1}{I(q)}} + {V_{pola}\exp^{- \frac{{({q - k})}\Delta\; t}{\tau}}} + {\sum\limits_{i = 1}^{q - k}\;{R_{2}{I\left( {i + k} \right)}\left\{ {\exp^{- \frac{{({{({q - k})} - i})}\Delta\; t}{\tau}} - \exp^{- \frac{{({{({q - k})} - i + 1})}\Delta\; t}{\tau}}} \right\}}}}} & \text{〈Equation~~1〉}\end{matrix}$

τ is a preset time constant of the RC pair.

FIG. 3 is a flowchart showing a method for estimating the resistance ofthe first resistor 210 as one of parameters of the equivalent circuitmodel 200 according to an embodiment of the present disclosure, and FIG.4 shows graphs for reference in describing the method of FIG. 3.

In step S310, the control unit 130 reads, from the memory 120,measurement data indicating a first number of terminal voltages and afirst number of currents measured by the sensing unit 110 in asequential order at each time step in a sliding time window having apredefined size. That is, the control unit 130 reads, from the memory120, the first number of terminal voltages and the first number ofcurrents recorded in the memory 120 over a predefined time in the pastfrom the current time step using the sliding time window of which theend time point has moved to the current time step. The predefined timeis equal to the size of the sliding time window. The first number is setby the predefined time and the time interval Δt between each time step.For example, when the predefined time=10 sec and Δt=0.01 sec, the firstnumber=10 sec/0.01 sec=1000. Each time the sliding time window moves byΔt, an oldest one of the first number of terminal voltages is discarded,and a newly measured terminal voltage is added. Likewise, each time thesliding time window moves by Δt, an oldest one of the first number ofcurrents is discarded, and a newly measured current is added.

The first number of terminal voltages include terminal voltage V(n)measured at the current time step and terminal voltage V(n−1) measuredat the previous time step. The first number of currents include currentI(n) measured at the current time step and current I(n−1) measured atthe previous time step.

In step S320, the control unit 130 calculates a voltage variation ΔV(n)of the current time step based on the terminal voltage V(n) measured atthe current time step and the terminal voltage V(n−1) measured at theprevious time step. In this instance, the control unit 130 may calculatethe voltage variation ΔV(n) by subtracting the terminal voltage V(n−1)measured at the previous time step from the terminal voltage V(n)measured at the current time step. That is, ΔV(n)=V(n)−V(n−1).

In step S330, the control unit 130 calculates a current variation ΔI(n)of the current time step based on the current I(n) measured at thecurrent time step and the current I(n−1) measured at the previous timestep. In this instance, the control unit 130 may calculate the currentvariation ΔI(n) by subtracting I(n−1) measured at the previous time stepfrom I(n) measured at the current time step. That is, ΔI(n)=I(n)−I(n−1).

Dissimilar to that of FIG. 3, the step S330 may be performed earlierthan the step S320, or at the same time as the step S320.

In step S340, the control unit 130 determines whether the voltagevariation ΔV(n) and the current variation ΔI(n) satisfy a first datafiltering condition. The first data filtering condition is a criterionfor determination as to whether ΔV(n) and ΔI(n) are suitable as learningdata for estimating the resistance of the first resistor 210.

When (i) the absolute value of the current variation ΔI(n) is largerthan a first threshold, and (ii) the multiplication of the voltagevariation ΔV(n) and the current variation ΔI(n) is larger than 0, thecontrol unit 130 may determine that the first data filtering conditionis satisfied.

The first threshold is a real number greater than 0, and is preset onthe basis of a measurement error of the current measuring unit 111. Thefirst resistor 210 is for modeling instantaneous voltage fluctuationsformed due to the internal resistance of the battery 10, therefore whenthe absolute value of ΔI(n) is larger than the first threshold, it issuitable to use ΔI(n) to estimate the resistance of the first resistor210 at the current time step. In contrast, when the absolute value ofΔI(n) is equal to or less than the first threshold, it is highly likelythat ΔI(n) results from the measurement error of the current measuringunit 111, and thus it is unsuitable to use ΔI(n) to estimate theresistance of the first resistor 210 at the current time step.

Additionally, according to the Ohm's law, the voltage of the firstresistor 210 is proportional to the current flowing through the firstresistor 210. Accordingly, only when ΔV(n) and ΔI(n) have the same sign,it is suitable to use ΔV(n) and ΔI(n) to estimate the resistance of thefirst resistor 210 at the current time step. In contrast, ΔV(n) having apositive value and ΔI(n) having a negative value or V(n) having anegative value and ΔI(n) having a positive value signify that a voltagechange of the first resistor 210 is against the Ohm's law, and thus itis unsuitable to use ΔI(n) to estimate the resistance of the firstresistor 210 at the current time step. Each of the two graphs shown inFIG. 4 shows a change in voltage and current of the battery 10 in thesame time range. In FIG. 4, the voltage and the current satisfying thefirst data filtering condition are each marked by a bold dot.

When the value of the step S340 is “YES”, the method moves to step S350.On the contrary, when the value of the step S340 is “NO”, the methodmoves to step S360.

In step S350, the control unit 130 estimates the resistance of the firstresistor 210 at the current time step based on the resistanceR_(1_est)(n−1) of the first resistor 210 estimated at the previous timestep, the voltage variation ΔV(n) and the current variation ΔI(n).

The control unit 130 may estimate the resistance of the first resistor210 at the current time step using the recursive least square (RLS)algorithm, and a detailed description will be provided below.

First, the weighted sum of squared errors S1 related to resistanceestimation of the first resistor 210 may be expressed as the followingEquation 2.

$\begin{matrix}{{S\; 1} = {\sum\limits_{k = 1}^{n}\;{\lambda^{n - k}\left\{ {{\Delta\;{V(k)}} - {{R_{1{\_{est}}}(n)}\Delta\;{I(k)}}} \right\}^{2}}}} & \text{〈Equation~~2〉}\end{matrix}$

In Equation 2, R_(1_est)(n) is the resistance of the first resistor 210to be estimated. Additionally, in Equation 2, λ is a first forgettingfactor which is preset as being greater than 0 and smaller than 1. λwill give a smaller influence on the resistance estimation of the firstresistor 210 as the terminal voltage and the current are measured at anearlier time in the past from the current time step.

The solution of the weighted sum of squared errors S1, i.e.,R_(1_est)(n) to minimize S1, may be calculated by the followingEquations 3 and 4.

$\begin{matrix}{{P_{1}(n)} = {\frac{1}{\lambda}\left\{ {{P_{1}\left( {n - 1} \right)} - \frac{{P_{1}\left( {n - 1} \right)}^{2}\Delta\;{I(n)}^{2}}{\lambda + {{P_{1}\left( {n - 1} \right)}\Delta\;{I(n)}^{2}}}} \right\}}} & \text{〈Equation~~3〉}\end{matrix}$R _(1_est)(n)=R _(1_est)(n−1)+P ₁(n)ΔI(n){ΔV(n)−R_(1_est)(n−1)ΔI(n)}  <Equation 4>

P₁(n) and P₁(n−1) are a correction factor of the current time step and acorrection factor of the previous time step, respectively. That is,P₁(n−1) is updated to P₁(n) by Equation 4.

In Equation 4, R_(1_est)(n−1) is the pre-estimated resistance of thefirst resistor 210 at the previous time step. The control unit 130 maycalculate the estimated resistance R_(1_est)(n) of the first resistor210 at the current time step using Equation 3 and Equation 4.

For the case in which a value of the symbol n indicating the currenttime step becomes 1 due to the initialization of the battery managementsystem 100, P₁(0) and R_(1_est)(0) may be pre-stored in the memory 120as different initial values. For example, P₁(0)=(1−λ)/(TH₁)², and TH₁may be equal to the first threshold. Additionally, R_(1_est)(0) may be apreset value corresponds to the temperature of the battery 10 measuredat the initial time step. The control unit 130 stores the estimatedresistance R_(1_est)(n) of the first resistor 210 at the current timestep in the memory 120.

In step S360, the control unit 130 sets the resistance R_(1_est)(n−1) ofthe first resistor 210 estimated at the previous time step as theresistance R_(1_est)(n) of the first resistor 210 estimated at thecurrent time step. That is, the resistance of the first resistor at thecurrent time step is handled as being equal to the resistanceR_(1_est)(n−1) of the first resistor 210 estimated at the previous timestep. Accordingly, dissimilar to S350, R_(1_est)(n)=R_(1_est)(n−1).

FIGS. 5 and 6 are flowcharts showing a method for estimating theresistance of the second resistor 220 as another one of parameters ofthe equivalent circuit model 200 according to an embodiment of thepresent disclosure, and FIG. 7 shows a graph for reference in describingthe method of FIG. 5.

In step S510, the control unit 130 determines whether the first numberof currents satisfy a second data filtering condition. The second datafiltering condition is a criterion for determination as to whether thefirst number of terminal voltages and the first number of currents aresuitable as learning data for resistance estimation of the secondresistor 220.

When a difference between the maximum and the minimum of the firstnumber of currents is larger than a second threshold, the control unit130 may determine that the second data filtering condition is satisfied.The graph shown in FIG. 7 shows a change in the current of the battery10 measured for a longer time than the size of the sliding time window.Assume that the size of the sliding time window is 10 sec, and thesecond threshold is 10 A. Seeing FIG. 7, a difference between themaximum and the minimum of the current measured from 330 sec to 340 secis 100 A or above. Accordingly, the current measured from 330 sec to 340sec satisfies the second data filtering condition. In contrast, thecurrent measured from 390 sec to 400 sec is constant, and does notsatisfy the second data filtering condition.

Due to the capacitor 230, the voltage of the second resistor 220 changesmore slowly than the voltage of the first resistor 210. Accordingly, itis preferred that the second threshold is larger than the firstthreshold.

When the value of the step S510 is “YES”, step S520 is performed. Whenthe value of the step S510 is “NO”, step S630 is performed.

In step S520, the control unit 130 generates a measured voltage vectorbased on the first number of terminal voltages and a measured currentvector based on the first number of currents. Hereinafter, assume thatthe first number is m of 2 or greater. Those skilled in the art willunderstand that n indicating the order of the current time step islarger than m.

The measured voltage vector may be expressed as m×1 matrix as below.V _(vec)=[V(n−m+1)V(n−m+2)V(n−m+3)V(n)]^(T)

The measured current vector may be expressed as m×1 matrix as below.I _(vec)=[I(n−m+1)I(n−m+2)I(n−m+3)I(n)]^(T)

In the above, the symbol T indicates the transposed matrix.

In step S530, the control unit 130 generates a reference voltage vectorbased on the measured voltage vector, the measured current vector andthe resistance R_(1_est)(n) of the first resistor 210 estimated at thecurrent time step. R_(1_est)(n) indicates the internal resistance of thebattery 10, and assume that R_(1_est)(n) is constant in the sliding timewindow. Then, the reference voltage vector may be expressed as below.Y _(w_vec) =V _(vec) −R _(1_est)(n)I _(vec)

The reference voltage vector Y_(w_vec) indicates a result of subtractingthe voltage of the internal resistance by each of the first number ofcurrents from each of the first number of terminal voltages.

In step S540, the control unit 130 may generate a first feature vector,a first parameter vector and a first model voltage vector based on theequivalent circuit model 200 and the measured current vector. The firstmodel voltage vector is the multiplication of the first feature vectorand the first parameter vector.

Let us r, K_(vec), H_(ind_vec) and 1_(vec) be each defined as below.

$r = {\exp\left( {- \frac{\Delta\; t}{\tau}} \right)}$$K_{vec} = \begin{bmatrix}r & r^{2} & r^{3} & \cdots & r^{m}\end{bmatrix}^{T}$$H_{{ind}\_{vec}} = {{\left( {1 - r} \right)\begin{bmatrix}1 & 0 & 0 & \cdots & 0 \\r & 1 & 0 & \cdots & 0 \\r^{2} & r & 1 & \cdots & 0 \\\vdots & \vdots & \vdots & \ddots & \vdots \\r^{m - 1} & r^{m - 2} & r^{m - 3} & \cdots & 1\end{bmatrix}}I_{vec}}$ $1_{vec} = \begin{bmatrix}1 & 1 & 1 & \ldots & 1\end{bmatrix}^{T}$

In case that the OCV of the voltage source 205 is constant as V_(ocv) inthe sliding time window, when the above definition is applied toEquation 1, the first model voltage vector expressed as the followingEquation 5 may be derived.V _(model1_vec) =V _(ocv)1_(vec) R ₂ H _(ind_vec) +V _(pola) K_(vec)=[1_(vec) H _(ind_vec) K _(vec)][V _(ocv) R ₂ V _(pola)]^(T)  <Equation 5>

In Equation 5, when X_(1_vec)=[1_(vec) H_(ind_vec) K_(vec)],β_(1_vec)=[V_(ocv)R₂V_(pola)]^(T), Equation 5 may be simplified as thefollowing Equation 6. The first model voltage vector may be a result ofmodeling the reference voltage vector.V _(model1_vec) =X _(1_vec)β_(1_vec)  <Equation 6>

X_(1_vec) is the first feature vector expressed as m×3 matrix. β_(1_vec)is the first parameter vector expressed as 3×1 matrix, and converts thefirst feature vector to the first model voltage vector. The threecomponents included in the first parameter vector are all unknown.

In step S550, the control unit 130 estimates the resistance of thesecond resistor 220 indicating the transient response history of thebattery in the sliding time window based on the reference voltage vectorand the first feature vector.

The sum of square errors S2 between the reference voltage vector and thefirst model voltage vector may be expressed as the following Equation 7.S2=∥Y _(w_vec) −V _(model1_vec)∥² =∥Y _(w_vec) −X_(1_vec)β_(1_vec)μ²  <Equation 7>

The control unit 130 may estimate the first parameter vector to minimizethe sum of square errors S2 using the following Equation 8.β_(1_win)=[V _(ocv_win) R _(2_win) V _(pola_win)]^(T)=(X _(1_vec) ^(T) X_(1_vec))⁻¹ X _(1_vec) ^(T) Y _(w_vec)   <Equation 8>

The transient response history of the battery generated in the slidingtime window is defined by the first number of terminal voltages and thefirst number of currents. Accordingly, the component R_(2_win) of thefirst parameter vector estimated using the above Equation 8 is theestimated resistance of the second resistor 220 indicating the transientresponse history of the battery in the sliding time window.

In step S560, the control unit 130 may calculate a first error valuecorresponding to the sum of least square errors S3 between the referencevoltage vector and the first model voltage vector based on the referencevoltage vector and the first feature vector.

The control unit 130 may calculate the sum of least square errors S3using the following Equation 9 related to the method of least squares.S3=Y _(w_vec) ^(T)(E−X _(1_vec)(X _(1_vec) ^(T) X _(1_vec))⁻¹ X _(1_vec)^(T))Y _(w_vec)   <Equation 9>

In Equation 9, E is the unit matrix.

The first error value may be any one of (i) S3, (ii) the mean of S3,i.e., S3/m, and (iii) the square root of S3/m.

In step S570, the control unit 130 may generate a second feature vector,a second parameter vector and a second model voltage vector based on asubstitute circuit model as a result of removing the second resistor 220from the equivalent circuit model 200 and the measured current vector.The second model voltage vector is the multiplication of the secondfeature vector and the second parameter vector.

As the substitute circuit model is free of the second resistor 220 ofthe equivalent circuit model 200, the second parameter vector isexpressed as β_(2_v)=[V_(ocv) V_(pola)]^(T) as a result of removing R₂from the first parameter vector, and the second feature vector isexpressed as X_(2_vec)=[lv kv] as a result of removing H_(ind_v) fromthe first feature vector. Accordingly, the second model voltage vectormay be expressed as the following Equation 10.V _(model2_vec) =X _(2_vec)β_(2_vec)  <Equation 10>

In step S580, the control unit 130 may calculate a second error valuecorresponding to the sum of least square errors S4 between the referencevoltage vector and the second model voltage vector based on thereference voltage vector and the second feature vector.

The control unit 130 may calculate the sum of least square errors S4using the following Equation 11 related to the method of least squares.S4=Y _(w_vec) ^(T)(E−X _(2_vec)(X _(2_vec) ^(T) X _(2_vec))⁻¹ X _(2_vec)^(T))Y _(w_vec)   <Equation 11>

The second error value may be any one of (i) S4, (ii) the mean of S4,i.e., S4/m, and (iii) the square root of S4/m.

In step S610, the control unit 130 determines whether a third datafiltering condition is satisfied based on the estimated resistanceR_(2_win) of the second resistor 220 indicating the transient responsehistory of the battery in the sliding time window, the first error valueand the second error value.

In detail, in step S610, the control unit 130 determines each of (i)whether R_(2_win) is larger than 0 ohm, and (ii) whether the seconderror value is larger than a value obtained by multiplying the firsterror value by a scaling factor (for example, 1.1) preset greaterthan 1. That is, the third data filtering condition may be satisfiedwhen R_(2_win) is larger than 0 ohm and the second error value is largerthan a value obtained by multiplying the first error value by thescaling factor.

In the physical aspect, actually, the resistance of the second resistor220 cannot be 0 ohm or less. Accordingly, R_(2_win) of 0 ohm or lessindicates that the first number of terminal voltages and the firstnumber of currents are unsuitable for resistance estimation of thesecond resistor 220. Additionally, as described above, the second errorvalue indicates that the polarization voltage by the RC pair is nottaken into consideration. Accordingly, the value obtained by multiplyingthe first error value by the scaling factor being larger than the seconderror value indicates that the first number of terminal voltages and thefirst number of currents fail to properly reflect the dynamiccharacteristics of voltage generated by the second resistor 220.

When the third data filtering condition is satisfied, S620 is performed,and otherwise, S630 is performed.

In step S620, the control unit 130 estimates the resistance of thesecond resistor 220 indicating the transient response history of thebattery in the current observation period based on the measured voltagevector, the measured current vector, the resistance R_(1_est)(n) of thefirst resistor 210 estimated at the current time step and the estimatedresistance R_(2_est)(n−1) of the second resistor 220 indicating thetransient response history of the battery 10 in the previous observationperiod. The previous observation period is a period from the initialtime step to the previous time step. The current observation period is aperiod from the initial time step to the current time step.

The control unit 130 may estimate the resistance of the second resistor220 at the current time step using the following Equation 12representing a function based on the RLS algorithm.R _(2_est)(n)=f(R _(1_est)(n),R _(2_est)(n−1),V _(vec) ,I_(vec))  <Equation 12>

In Equation 12, the function f( ) outputs R_(2_est)(n) whenR_(1_est)(n), R_(2_est)(n−1), V_(vec) and I_(vec) are inputted.R_(2_est)(n−1) is the estimated resistance of the second resistor 220indicating the transient response history of the battery in the previousobservation period. Likewise, R_(2_est)(n) is the estimated resistanceof the second resistor 220 indicating the transient response history ofthe battery in the current observation period.

In step S630, the control unit 130 sets the estimated resistanceR_(2_est)(n−1) of the second resistor 220 indicating the transientresponse history of the battery 10 in the previous observation period asthe estimated resistance R_(2_est)(n) of the second resistor 220indicating the transient response history of the battery 10 in thecurrent observation period. That is, the transient response history ofthe battery 10 in the previous observation period is handled as beingequal to the transient response history of the battery 10 in the currentobservation period. Accordingly, dissimilar to S620,R_(2_est)(n)=R_(2_est)(n−1).

The control unit 130 may predict the terminal voltage of the battery 10using the estimated resistance R_(1_est)(n) of the first resistor 210and the estimated resistance R_(2_est)(n) of the second resistor 220,and adjust the duty cycle of the switching signal outputted to theswitch 20 based on the predicted terminal voltage.

The control unit 130 may estimate the SOC of the battery 10 at thecurrent time step using the estimated resistance R_(1_est)(n) of thefirst resistor 210 and the estimated resistance R_(2_est)(n) of thesecond resistor 220, and adjust the duty cycle of the switching signaloutputted to the switch 20 based on the estimated SOC.

FIG. 8 is a flowchart showing a method for determining the charge powerlimit and the discharge power limit of the battery 10 according toanother embodiment of the present disclosure, and FIGS. 9 and 10 showdifferent exemplary graphs for reference in describing the method ofFIG. 8. The method of FIG. 8 starts after the resistance of the firstresistor 210 and the resistance of the second resistor 220 are estimatedthrough the method of FIGS. 3, 5 and 6.

Referring to FIG. 8, in step S800, the control unit 130 may estimate thepolarization voltage V_(pola)(n) of the battery 10. The polarizationvoltage V_(pola)(n) may be estimated through a variety of known methods.The control unit 130 may use V_(pola_win) estimated using Equation 8 inthe step S550 of FIG. 5 as the polarization voltage V_(pola)(n), and inthis case, the step S800 may be omitted.

In step S810, the control unit 130 predicts the terminal voltage of thebattery 10 after a predefined time T_(hard) based on the resistanceR_(2_est)(n) of the second resistor 220, the measured terminal voltageV(n), the measured current I(n) and the polarization voltage V_(pola)(n)of the battery 10.

Assume that the current of the battery 10 is constant as hard for thepredefined time T_(hard) from the current time step. I_(hard) may beequal to or larger or smaller than I(n). When it is assumed that the OCVof the battery 10 is constant for the predefined time T_(hard) from thecurrent time step, the terminal voltage V_(t) of the battery 10 at anarbitrary time point t in the period until the predefined time T_(hard)has passed from the current time step may be expressed as the followingEquation 13.

$\begin{matrix}{V_{t} = {{V(n)} + {\left\{ {1 - \exp^{- \frac{t - t_{n}}{\tau}}} \right\}\left\{ {{{R_{2{\_{est}}}(n)}{I(n)}} - {V_{pola}(n)}} \right\}} + {\left\{ {{R_{1{\_{est}}}(n)} + {\left\{ {1 - \exp^{- \frac{t - t_{n}}{\tau}}} \right\}{R_{2{\_{est}}}(n)}}} \right\}\left\{ {I_{hard} - {I(n)}} \right\}}}} & \text{〈Equation~~13〉}\end{matrix}$

t_(n) is the time point indicating the current time step.

The control unit 130 may calculate the predicted terminal voltage usingthe following Equation 14. Equation 14 may be derived by substitutingI(n) into I_(hard) of Equation 13 and t_(n)+T_(hard) into t.

$\begin{matrix}{{V_{pred}(n)} = {{V(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}\left\{ {{{R_{2{\_{est}}}(n)}{I(n)}} - {V_{pola}(n)}} \right\}}}} & \text{〈Equation~~14〉}\end{matrix}$

V_(pred)(n) is the predicted terminal voltage.

In step S820, the control unit 130 predicts the net resistance of thebattery 10 in the predefined time T_(hard) based on the resistanceR_(1_est)(n) of the first resistor 210 and the resistance R_(2_est)(n)of the second resistor 220. The control unit 130 may calculate thepredicted the net resistance using the following Equation 15.

$\begin{matrix}{{R_{net}(n)} = {{R_{1{\_{est}}}(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}{R_{2{\_{est}}}(n)}}}} & \text{〈Equation~~15〉}\end{matrix}$

R_(net)(n) is the predicted net resistance. As described above,R_(1_est)(n) is larger than 0 ohm, thus R_(net)(n) is larger thanR_(1_est)(n).

In step S840, the control unit 130 determines the charge current limitI_(limit_c)(n). In detail, the control unit 130 determines the chargecurrent limit I_(limit_c)(n) based on the measured current I(n), thepredicted terminal voltage V_(pred)(n), the predicted net resistanceR_(net)(n) and the given maximum charge voltage V_(max). The controlunit 130 may determine the charge current limit I_(limit_c)(n) using thefollowing Equation 16.

$\begin{matrix}{{I_{{limit}\_ c}(n)} = {{I(n)} + \frac{V_{\max} - {V_{pred}(n)}}{R_{net}(n)}}} & \text{〈Equation~~16〉}\end{matrix}$

When I_(limit_c)(n) is larger than the given maximum charge currentI_(max), the control unit 130 may determine the charge current limitI_(limit_c)(n) as a same value as the given maximum charge currentI_(max).

In step S850, the control unit 130 determines the charge voltage limitV_(limit_c)(n). In detail, the control unit 130 determines the chargevoltage limit V_(limit_c)(n) based on the measured terminal voltageV(n), the measured current I(n), the charge current limit I_(limit_c)(n)and the resistance R_(1_est)(n) of the first resistor 210. The controlunit 130 may determine the charge voltage limit V_(limit_c)(n) using thefollowing Equation 17.V _(limit_c)(n)=V(n)+{I _(limit_c)(n)−I(n)}R _(1_set)(n)  <Equation 17>

In Equation 17, V_(limit_c)(n) is equal to t=t_(n) andI_(hard)=I_(limit_c)(n) of Equation 13.

FIG. 9 shows a voltage graph showing a change in terminal voltage of thebattery 10 when the current of the battery 10 is constant asI_(limit_c)(n) from t_(n) to t_(n)+T_(hard). That is, the voltage graphof FIG. 9 is obtained by substituting I_(limit_c)(n) into hard ofEquation 13. Referring to FIG. 9, in Equation 17, the use of V(n) andR_(1_est)(n) in place of V_(pred)(n) and R_(net)(n) is to determine thelowest voltage in the period for the predefined time T_(hard) from thecurrent time step as the charge voltage limit V_(limit_c)(n).

In step S860, the control unit 130 determines a state of power (SOP)indicating the charge power limit of the battery 10 based on the chargecurrent limit I_(limit_c)(n) and the charge voltage limitV_(limit_c)(n). The charge power limit is power that the battery 10 canbe supplied for the predefined time T_(hard) from the current time withthe charge current limit I_(limit_c)(n) and the charge voltage limitV_(limit_c)(n), and the charge power limit is equal to themultiplication of the charge current limit I_(limit_c)(n) and the chargevoltage limit V_(limit_c)(n).

In step S870, the control unit 130 determines the discharge currentlimit I_(limit_d)(n). In detail, the control unit 130 determines thedischarge current limit I_(limit_d)(n) based on the measured currentI(n), the predicted terminal voltage V_(pred)(n), the predicted netresistance R_(net)(n) and the given minimum discharge voltage V_(mim).The control unit 130 may determine the discharge current limitI_(limit_d)(n) using the following Equation 18.

$\begin{matrix}{{I_{{limit}\_ d}(n)} = {{I(n)} + \frac{V_{\min} - {V_{pred}(n)}}{R_{net}(n)}}} & \text{〈Equation~~18〉}\end{matrix}$

When I_(limit_d)(n) is smaller than the given minimum discharge currentI_(mim), the control unit 130 may determine the discharge current limitI_(limit_d)(n) as a same value as the given minimum discharge currentI_(mim).

In step S880, the control unit 130 determines the discharge voltagelimit V_(limit_d)(n). In detail, the control unit 130 determines thedischarge voltage limit V_(limit_d)(n) based on the measured terminalvoltage V(n), the measured current I(n), the discharge current limitI_(limit_d)(n) and the resistance R_(1_est)(n) of the first resistor210. The control unit 130 may determine the discharge voltage limitV_(limit_d)(n) using the following Equation 19.V _(limit_d)(n)=V _(pred)(n)+{I _(limit_d)(n)−I(n)}R_(net)(n)  <Equation 19>

In Equation 19, V_(limit_d)(n) is equal to t=t_(n)+T_(hard) andI_(hard)=I_(limit_d)(n) of Equation 13.

FIG. 10 shows a voltage graph showing a change in terminal voltage ofthe battery 10 when the current of the battery 10 is constant asI_(limit_d)(n) from the time point to indicating the current time stepto t_(n)+T_(hard). That is, the voltage graph of FIG. 10 is obtained bysubstituting I_(limit_d)(n) into I_(hard) of Equation 13. Referring toFIG. 10, in contrast with Equation 17, in Equation 19, the use ofV_(pred)(n) and R_(net)(n) in place of V(n) and R_(1_est)(n) is todetermine the lowest voltage in the period for the predefined timeT_(hard) from the current time step as the discharge voltage limitV_(limit_d)(n).

In step S890, the control unit 130 determines a SOP indicating thedischarge power limit of the battery 10 based on the discharge currentlimit I_(limit_d)(n) and the discharge voltage limit V_(limit_d)(n). Thedischarge power limit is power that the battery 10 can supply for thepredefined time T_(hard) from the current time with the dischargecurrent limit I_(limit_d)(n) and the discharge voltage limitV_(limit_d)(n), and is equal to the multiplication of the dischargecurrent limit I_(limit_d)(n) and the discharge voltage limitV_(limit_d)(n).

The control unit 130 may transmit a notification signal indicating thecharge voltage limit V_(limit_c)(n) and/or the discharge voltage limitV_(limit_d)(n) to the external device 2.

Data indicating the results of performing each step shown in FIGS. 3, 5,6 and 8 may be stored in the memory 120 by the control unit 130 whenevereach step is finished.

The embodiments of the present disclosure described hereinabove are notimplemented only through the apparatus and method, and may beimplemented through programs that perform functions corresponding to theconfigurations of the embodiments of the present disclosure or recordingmedia having the programs recorded thereon, and this implementation maybe easily achieved by those skilled in the art from the disclosure ofthe embodiments previously described.

While the present disclosure has been hereinabove described with regardto a limited number of embodiments and drawings, the present disclosureis not limited thereto and it is obvious to those skilled in the artthat various modifications and changes may be made thereto within thetechnical aspects of the present disclosure and the equivalent scope ofthe appended claims.

Additionally, as many substitutions, modifications and changes may bemade to the present disclosure described hereinabove by those skilled inthe art without departing from the technical aspects of the presentdisclosure, the present disclosure is not limited by the above-describedembodiments and the accompanying drawings, and some or all of theembodiments may be selectively combined to allow various modifications.

LIST OF REFERENCE NUMBERS

-   -   1: battery pack    -   10: battery    -   20: switch    -   100: battery management system    -   110: sensing unit    -   120: memory    -   130: control unit    -   140: communication interface    -   200: equivalent circuit model    -   210: first resistor    -   220: second resistor    -   230: capacitor

What is claimed is:
 1. A method for determining a power limit of abattery using an equivalent circuit model for the battery, theequivalent circuit model including a first resistor, a second resistorconnected in series to the first resistor, and a capacitor connected inparallel to the second resistor, the method comprising: measuring aterminal voltage and a current of the battery; estimating a resistanceof the first resistor, a resistance of the second resistor, and apolarization voltage of the battery; predicting a terminal voltage ofthe battery after a predefined time by inputting the resistance of thesecond resistor, the measured terminal voltage, the measured current,and the polarization voltage into a given Equation 1; predicting a netresistance of the battery after the predefined time by inputting theresistance of the first resistor and the resistance of the secondresistor into a given Equation 2; determining a charge current limit tobe equal to a sum of (i) the measured current and (ii) a differencebetween a given maximum charge voltage and the predicted terminalvoltage divided by the predicted net resistance; determining a chargevoltage limit to be equal to a sum of (i) the measured terminal voltageand (ii) a difference between the charge current limit and the measuredcurrent multiplied by the resistance of the first resistor; determininga charge power limit of the battery to be equal to the charge currentlimit multiplied by the charge voltage limit; and limiting one or bothof a charge voltage and a charge power of the battery based on thedetermined charge voltage limit and the determined charge power limit.2. The method according to claim 1, further comprising: determining adischarge current limit based on the measured current, the predictedterminal voltage, the predicted net resistance, and a given minimumdischarge voltage; determining a discharge voltage limit based on thepredicted terminal voltage, the measured current, the discharge currentlimit, and the net resistance; and determining a discharge power limitof the battery based on the discharge current limit and the dischargevoltage limit.
 3. The method according to claim 1, wherein the givenEquation 1 is as follows: $\begin{matrix}{{{V_{pred}(n)} = {{V(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}\left\{ {{{R_{2{\_{est}}}(n)}{I(n)}} - {V_{pola}(n)}} \right\}}}},} & \text{〈Equation~~1〉}\end{matrix}$ where: V(n) is the measured terminal voltage, R_(2_est)(n)is the estimated resistance of the second resistor, I(n) is the measuredcurrent, V_(pola)(n) is the polarization voltage, τ is a preset timeconstant of the second resistor and the capacitor, T_(hard) is thepredefined time, and V_(pred)(n) is the predicted terminal voltage. 4.The method according to claim 1, wherein the given Equation 2 is asfollows: $\begin{matrix}{{{R_{net}(n)} = {{R_{1{\_{est}}}(n)} + {\left\{ {1 - \exp^{- \frac{T_{hard}}{\tau}}} \right\}{R_{2{\_{est}}}(n)}}}},} & \text{〈Equation~~2〉}\end{matrix}$ where: R_(1_est)(n) is the estimated resistance of thefirst resistor, R_(2_est)(n) is the estimated resistance of the secondresistor, τ is a preset time constant of the second resistor and thecapacitor, T_(hard) is the predefined time, and R_(net)(n) is thepredicted net resistance.
 5. The method according to claim 1, whereinthe determining the charge current limit uses the following Equation 3:$\begin{matrix}{{{I_{{limit}\_ c}(n)} = {{I(n)} + \frac{V_{\max} - {V_{pred}(n)}}{R_{net}(n)}}},} & \text{〈Equation~~3〉}\end{matrix}$ where: I(n) is the measured current, R_(net)(n) is thepredicted net resistance, V_(pred)(n) is the predicted terminal voltage,V_(max) is the given maximum charge voltage, and I_(limit_c)(n) is thecharge current limit.
 6. The method according to claim 1, wherein thedetermining the charge voltage limit uses the following Equation 4:V _(limit_c)(n)=V(n)+{I _(limit_c)(n)−I(n)}R _(1_set)(n),  <Equation 4>where: V(n) is the measured terminal voltage, I(n) is the measuredcurrent, I_(limit_c)(n) is the charge current limit, R_(1_est)(n) is theestimated resistance of the first resistor, and V_(limit_c)(n) is thecharge voltage limit.
 7. The method according to claim 2, wherein thedetermining the discharge current limit uses the following Equation 5:$\begin{matrix}{{{I_{{limit}\_ d}(n)} = {{I(n)} + \frac{V_{\min} - {V_{pred}(n)}}{R_{net}(n)}}},} & \text{〈Equation~~5〉}\end{matrix}$ where: I(n) is the measured current, R_(net)(n) is thepredicted net resistance, V_(pred)(n) is the predicted terminal voltage,V_(min) is the given minimum charge voltage, and I_(limit_d)(n) is thedischarge current limit.
 8. The method according to claim 2, wherein thedetermining the discharge voltage limit uses the following Equation 6:V _(limit_d)(n)=V _(pred)(n)+{I _(limit_d)(n)−I(n)}R_(net)(n),  <Equation 6> where: V_(pred)(n) is the predicted terminalvoltage, I(n) is the measured current, I_(limit_d)(n) is the dischargecurrent limit, R_(net)(n) is the predicted net resistance, andV_(limit_d)(n) is the discharge voltage limit.
 9. A battery managementsystem for determining power limit of a battery using an equivalentcircuit model for the battery, the equivalent circuit model including afirst resistor, a second resistor connected in series to the firstresistor, and a capacitor connected in parallel to the second resistor,the battery management system comprising: a sensing unit configured tomeasure a terminal voltage and a current of the battery; and a controlunit operably coupled to the sensing unit, the control unit beingconfigured to: estimate a resistance of the first resistor, a resistanceof the second resistor, and a polarization voltage of the battery,predict a terminal voltage of the battery after a predefined time byinputting the resistance of the second resistor, the measured terminalvoltage, the measured current, and the polarization voltage into a givenEquation 1, predict a net resistance of the battery after the predefinedtime by inputting the resistance of the first resistor and theresistance of the second resistor into a given Equation 2; determine acharge current limit to be equal to a sum of (i) the measured currentand (ii) a difference between a given maximum charge voltage and thepredicted terminal voltage divided by the predicted net resistance;determine a charge voltage limit to be equal to a sum of (i) themeasured terminal voltage and (ii) a difference between the chargecurrent limit and the measured current multiplied by the resistance ofthe first resistor; determine a charge power limit of the battery to beequal to the charge current limit multiplied by the charge voltagelimit; and transmit a message to an external device to limit one or bothof a charge voltage and a charge power of the battery based on thedetermined charge voltage limit and the determined charge power limit.10. The battery management system according to claim 9, wherein thecontrol unit is further configured to: determine a discharge currentlimit based on the measured current, the predicted terminal voltage, thepredicted net resistance, and a given minimum discharge voltage;determine a discharge voltage limit based on the predicted terminalvoltage, the measured current, the discharge current limit, and the netresistance; and determine a discharge power limit of the battery basedon the discharge current limit and the discharge voltage limit.